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E4.7 Bretherton’s equation
The equation that we will discuss is

where is defined in and we will further assume that
we have suitable initial data for the type of solution that we seek. The aim is to produce
a solution, via the method of multiple scales, for First we observe that, with
there is a solution

where, given k (the wave number), (the frequency) is defined by the dispersion relation:

The presence of the parameter, together with a naïve asymptotic solution (generating
terms proportional to or suggests that we must expect changes on the slow
scales and The fast scale is defined in much the same way that we adopted for
E4.3 and E4.5; thus we write

The solution now sits in a domain in 3-space, defined by
A solution described by these variables will have the property that
both the wave number and the frequency slowly evolve. (Note that the correct form of
the solution with is recovered if k = constant and The retention
of the parameter in the definitions of k and allows us to treat these functions as
asymptotic expansions, if that is useful and relevant; often this is unnecessary.
From (4.61) we have the operator identities

and also

etc., as far as It is sufficient, for the results that we present here, to
transform equation (4.58) but retain terms no smaller than thus, with